[Smt-talk] Classical Form and Recursion

Dmitri Tymoczko dmitri at Princeton.EDU
Thu Apr 2 10:08:44 PDT 2009


I suspect we're nearing that magical limit where one person's  
evidence is another person's reductio, so I'll be brief.

> Owing to this property, music has, in my view, much stronger  
> potential for extensive recursive (prolongational) structuring than  
> has language.

I'll just report that every psychologist, linguist, and evolutionary  
biologist I've talked to about this issue has told me that they find  
this suggestion implausible.  They argue that language is so central  
to human survival that it would be extraordinary if our capacities  
for perceiving recursion were not utilized to their fullest extent.   
Consequently, they typically use limits on recursion-perception in  
language as a rough guide to limits on recursion-perception in  
music.  And we know that humans sometimes do quite badly at  
perceiving recursion in language: the sentence "People dogs cats like  
hate suck" is very difficult to parse, and it involves only three  
levels of center embedding.  More generally typical sentence lengths  
are on the order of 10 seconds, more than an order of magnitude  
shorter than typical tonal pieces.

> Beginning and ending the sentence with the same word plays no role  
> for syntactic closure in language. In your example sentence, the  
> subject happens to be the same as the object, but this coincidence  
> has no significance for syntax (only for semantics and rhetoric).  
> In tonal music, by contrast, there is a norm that closed harmonic  
> progressions begin and end with I (I hope you will agree that there  
> is such a norm). If a phrase starts on I and proceeds to other  
> harmonies, we are expecting a convincing return to I until this  
> happens. (If our expectations are not fulfilled and the phrase does  
> not return to I, we do not hear it as closed phrase, but await  
> continuation.) This demonstrates that the referential status of a  
> single element (tonic chord in this case) may have significance for  
> musical syntax in a way that differs fundamentally from that of a  
> single word for linguistic syntax. The perception of the syntax in  
> a tonal progression may be governed by an element in that  
> progression in a sense for which there is no linguistic  
> counterpart. (Closed tonic-to-tonic progressions are by no means  
> the only way to acheive such governing status, but they are a prime  
> example.)

My question was: what justifies the "reduction" of ABA to A.  Your  
answer seems to be "there is a norm that harmonic progressions begin  
and end with I."  But the existence of the norm is clearly  
insufficient.  Sumo wrestling matches begin and end with bowing.  The  
bowing is  a time of lowered tension, relative to the match.  But  
nobody takes the wrestling to "represent" or "prolong" the bowing.

I don't mean this to be flippant.  What I want to know is what  
principles justify the "reduction" of ABA to A.  I think this is a  
serious, interesting question.  Many of the answers that are commonly  
proffered -- "we have a norm of ABA structuring," or "A is less tense  
than B" -- are clearly insufficient.  The example of the Sumo  
wrestlers is silly in one sense, but it dramatizes the fact that  
these answers offered don't do the job.  Whether you think that's  
important or not probably depends on whether you're convinced that  
there *is* a good underlying justification somewhere (as you are) or  
whether (like me) you are agnostic or skeptical about the issue.

> For testing whether a listener actually perceives tonal closure in  
> m. 3, one might consider the following experiment, though it has a  
> deficiency. Listen to the progression (1) as written above and (2)  
> as a truncated version, breaking of after bar 2, beat 2. If one  
> finds (1) embodying more convincing closure than (2), this speaks  
> to prolongational perception. The deficiency in this experiment is  
> that (2) does not include all the information that supports  
> perceiving bar 2, beat 2 as subordinate to the surrounding  
> dominant, since part of this information comes retrospectively  
> through the return of V (2^) at beat 3. Nevertheless, even without  
> this retrospective information, I find (2) less satisfactory than  
> (1) in terms of closure.

The challenge is to control for the nonrecursive facts that  
contribute to closure.  It's really a very difficult problem; just  
truncating progressions won't do the job, as we may have an  
expectation that phrases last a certain length.


Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)

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